The decision variables and the objective function

A company that has a two-year contract to haul ore from an open-pit mine to loading docks for shipping needs 200 additional trucks. Purchased trucks have a useful life of two years and a purchase cost of $140,000 each. The company can lease trucks for $80,000 per year (paid at the beginning of the year). Purchased trucks will be purchased only at the beginning of the two-year period and have no salvage value at the end of two years. The mining company has $8 million cash available to lease and/or buy trucks at the beginning of year 1. In addition, the company can obtain a loan each year for as mush as $20 million at 12% interest per year. The loan agreement requires that the company repay the borrowed
amount plus interest at the end of each year. Each truck will earn $120,000 per year, which will become part of the cash flow of funds available to the company for truck leasing and loan repayment. The company wants to minimize the total cost of the trucks over the two-year period. Formulate a linear programming model for this problem. (Do not attempt to solve.)

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The decision variables and the objective function have to be defined. For this, look for the sentence in the problem statement that mentions the goal. Usually this sentence will be towards the end of the problem statement. Here it is "The company wants to minimize the total cost of the trucks over the two-year period". So the objective is to minimize the total cost of trucks. Now we must determine the decision variable. Each variable will represent a different alternative that can be chosen, so we look for sentences in the problem statement which talk about alternatives. Here: "Purchased trucks ... The company can lease trucks...". So the choice is between purchasing trucks and leasing trucks. The purchased trucks will be used for two years, whereas the leased trucks will be used for one year after they're leased. This means some trucks must be leased at the start for the first year, and at the start of the second year, their lease will expire and more trucks will have to be leased. Since the number of purchased trucks does not change, the same number of trucks must be leased at the start of the second year, as at the start of the first year. So we have ...